{
  "abstract": "Let $G$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ assigns to each vertex and each edge a label from $\\{1,\\ldots,k\\}$. The weight of a vertex is the sum of its label and the labels of its incident edges. A total labeling is vertex irregular if all vertex weights are distinct. The total vertex irregularity strength $\\text{tvs}(G)$ is the smallest $k$ for which $G$ has a vertex irregular total $k$-labeling. For an $r$-regular graph $G$ on $n$ vertices, a counting argument gives $\\text{tvs}(G)\\ge\\lceil(n+r)/(r+1)\\rceil$. The restriction of a conjecture of Nurdin, Baskoro, Salman, and Gaos to regular graphs asserts that this bound is attained. We prove this assertion for cubic and $4$-regular graphs. We also show that, for every fixed $r\\ge2$, a recent theorem on prescribed degree frequencies implies the assertion for all sufficiently large $r$-regular graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30114",
  "author_total": 2,
  "authors": [
    {
      "name": "Songling Shan"
    },
    {
      "name": "Yucheng Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fk8twhf506va1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30114",
  "fulltext_url": null,
  "id": "fk8twhf506va1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30114",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7214334114",
  "pdf_url": "https://arxiv.org/pdf/2609.30114",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fk8twhf506va1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30114",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30114"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Total Vertex Irregularity Strength of Cubic and 4-Regular Graphs",
  "title_text": "Total Vertex Irregularity Strength of Cubic and 4-Regular Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/fk8twhf506va1",
  "venue": "arXiv",
  "versions": []
}